Proving Cauchy Sequence of |an-bn|

[rbzima]

I'm basically trying to show that if (a_n) and (b_n) are Cauchy sequences, then (c_n) = |a_n - b_n| is also a Cauchy sequence.

I know that the triangle inequality is going to be used at one point or another, but I suppose I'm a little confused because:

(a_n) is Cauchy implies |a_n - a_m| < e
(b_n) is Cauchy implies |b_n - b_m| < e

I think at some point my e's are going to be changed to e/2, which is totally legitimate because e is arbitrary anyway.

 

[mathman]

The key to this is to show that ||X|-|Y||<=|X-Y|

Then ||a_n-b_n|-|a_m-b_m||<=|(a_n-a_m)-(b_n-b_m)|<=|a_n-a_m|+|b_n-b_m|